Looks like a storm door mounted inside rather than out to me. Don't know how much you want to spend but the Larson door with a hidden spring and soft close is nice but it mounts about 1" away from the jamb rather than bring flush (if that makes sense).
Have you searched "in swing storm door handle". Seems like that swing style latch should be easy to find.
https://www.swisco.com/In-swinging-Door-Latch/pd/Replacement-Storm-Door-Handles_Locks/40-028?gad_source=1&gbraid=0AAAAAD_HAIi60H1Bm7laMd-SDaSEGGGwn&gclid=CjwKCAjwq7fABhB2EiwAwk-YbMayJ6Tr9A9-qOH-YuqQkzHx4-bNPQQ9XuroZvoZDN-7aOFdIeHPvRoCKHYQAvD_BwE
Don't know why that amazon link didn't work. Put this in an Amazon search bar "Screen and Storm Door Latch, Non-Keyed, Surface Mount, for Inswinging Doors, Aluminum"
I can't find an actual image of this one but this seems like it would work and has a lock. Brixwell 900-7323